The Gray tensor product of 2-categories $C$ and $D$ is a "fattening up" of the cartesian product $C\times D$ in which the equality $(f,1)(1,g) = (1,g)(f,1)$ is replaced by a 2-morphism. Nowadays the term "Gray tensor product" more often seems to refer to the pseudo version in which these 2-morphisms are invertible, but Gray's original version was "lax" (or colax) and had them not necessarily invertible.

where $\mathrm{Fun}_w$ denotes the 2-category of strict 2-functors and $w$-natural transformations. One can also spell out explicitly the morphisms which are represented by $C\otimes_w D$ as a sort of "$w$-cubical functor"; these can be identified with a certain class of $w$-functors $C\times D\to E$ which are strict in certain ways.

In sum, the Gray tensor product is a beautiful thing for talking about strict 2-functors and all sorts of weak natural transformations. My question is, what happens when we move to pseudo 2-functors? I'm happy to keep my 2-categories strict and not to worry about lax or oplax 2-functors. Is there an equivalence of bicategories

where ${\mathrm{PsFun}_w (-,-)}$ denotes the 2-category of pseudofunctors and $w$-natural transformations? This is true when $w=$ pseudo, since in that case $C\otimes D$ is equivalent to $C\times D$ as a bicategory, and the tricategory of bicategories is cartesian closed with internal-hom $\mathrm{PsFun}_{\mathrm{pseudo}}$. But what about when $w=$ lax?

See JW Gray Formal Category Theory: Adjointness for 2-Categories (Lnm 391) p. 86. I guess that this has a formulation in terms of tricategories, and $lax_{(3)}$-funtors
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Buschi SergioMay 11 '12 at 11:42

@Buschi: Gray uses "pseudo-functor" to mean what nowadays is called a "lax functor". As far as I can tell, his discussion of why the tensor product doesn't work for lax functors has no bearing on pseudo-functors.
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Mike ShulmanMay 11 '12 at 18:28

@Mike: Yup, this is basically true and can be proved using the folk model structure and showing that the lax tensor product is left-derivable in its second component.
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Harry GindiJun 23 '12 at 9:36

@Harry: Sounds like a plausible approach... but you sound very confident, have you worked out the details?
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Mike ShulmanJun 25 '12 at 15:46

1 Answer
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Consider at first normal pseudo-funtors (on 2-cetegories) we call a pseudo.functor $F: \mathcal{A}\to\mathcal{B}$ normal if for any $A$: $F(1_A)=1_{FA}$ and the canonical isomorphism is the identity. Let $Fun_{np}(\mathcal{A}, \mathcal{B})$ the category of normal pseudofuntors and lax-transformations (with modifications too, is a 2-category). Now a normal pseudfunctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$ give a family of normal pseudofunctors

as in the diagram of [G] p. 57, which verify the properties $QF_21, QF_22,\ QF_23$ of of [G] p. 57. We call this "data" a normal quasi-pseudo-funtor.
Similarly a 2-cell to induce what is called a lax transformation between normal quasi-pseudo-funtor.

mutually this data describe exactly a normal pseudofuctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$ (this is sketched in [G] p. 60 for 2-functors) and this involve also lax-transformation, then we have a (isomorphism):

Similarly we define $\phi'_{f, g}$ also if other of the objects $A,\ B, C$ are of type $I_D$ for some object $D$.

remains the verification of the conditions of consistency, but this follow from the general criterion of coherency for pseudo-functors (S. MacLane, R. Paré, "Coherence for bicategories and indexed categories")
or for direct verification

I don't understand your edit: do you mean it doesn't work in the non-normal case?
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Mike ShulmanJun 22 '12 at 20:27

THe edit mean that I'm working about non-normal case I actually missed the problem because it seemed that nobody cared. I know how to go forward, if you are interested I'll go ahead. (sorry for my English)
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Buschi SergioJun 23 '12 at 5:57